The Bernoulli Equation
Patrick Bruskiewich, M.Sc.
Department of Physics and Astronomy,
University of British Columbia
6224 Agricultural Road,
Vancouver, B.C., Canada V6T 1Z1
A
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4,
Bernoulli's Principle: where the velocity of an
incompressible fluid (gas or liquid) is high, the
pressure is low and where the velocity is low,
the pressure is high.
Bernoulli's Principle makes sense: Fluids will
flow from a region of higher pressure to a region
of lower pressure and in doing so increase their
speed as a result.
If this were not the case => if the pressure were
higher at a point of higher velocity then the
fluid would be slowed down at that point. As
a result this would be a point of lower velocity.
This is a contradiction!
?
Bernouilli developed an equation of state that
expresses his principle quantitatively. Assume
the fluid flow is steady and incompressible.
fe£ " v
Volume ~V,
V^tavwe r V*
f77777^'
Consider the work to move a small volume V\
of fluid at point 1,
W 1 = F x Ah = P 1 A 1 Al 1
(1)
At point 2 the force exerted on the fluid is
opposite to the motion so then
W 2 = - F 2 Al 2 = - P 2 A 2 Al 2
(2)
3
Work is also done on the fluid to counter the
force of gravity.
The work done to lift the mass of fluid in volume
Vi from point 1 to point 2 is
W g = - mg(y 2 - y\) (3)
So then the net work W net done on the fluid in
moving it from point 1 to point 2 is and is given
by
W net = W 1 + W 2 + Wg
= P 1 A 1 Al 1 - P 2 A 2 Al 2 - mg(y 2 -yi) (4)
According to the work-energy principle, the net
work on a system is equal to its change in kinetic
energy. That is
1 1
W net = ARE — -mv 2 - -mvi (5)
So then
1 9 1 o
-rnv 2 - -mv 1 = P 1 A 1 Al 1 - P 2 A 2 Al 2 - mg(y 2 - Vi)
(6)
Gather the 1 and 2 terms together we find
1 2
P 2 A 2 Al 2 + -mv 2 + mgy 2
i t~\
PlA 1 Ali + - mv l + m 9V\
Z
R
We can now factor out the volume on each
side of the equation, remembering that p = y t
namely
1 2
A 2 AZ 2 (P 2 + ~pv 2 + P9V2)
1 2
= AiAZi(Pi + -pv 1 + P^yi)
(7)
We know from the continuity equation that A 2 AZ 2
= AiAZi, which is the volume of the fluid ele-
ment flowing through the system.
We can therefore cancel these terms off both
sides of the equation.
What we are left with is
1 2
P 2 + ~pv 2 + P9V2
Pi + ~pv{ + pgy ±
This is Bernoulli's equation
Since points 1 and 2 can be any points along a
tube of flow, Bernouilli's equation can be writ-
ten
1 2
p 2 + ~P v 2 + P 9 V2 = constant (8)
a.
at every point in the fluid where y is the height
of the centre of the tube above the reference
level.
The Torrjcelli Apparatus
b* %-y t
v;*o
%x -r
r,
AX
H
The top of the apparatus and the spigot are
open to the atmosphere so that J\ = P 2 =
*atm ■
Using Bernoulli's equation (> 2 « 0) we have
~P v i + P 9 Vi = P 9 V2
(9)
Solving for v\ we find
,2 _
vf = 2g(v2 - vi) (io)
so that
vi = \][2g(y 2 - vi)] (11)
This is known as Torricelli's theorem, which was
discovered about 100 years before the Bernoulli
equation was developed.
Torricelli was a student of Galileo Galilei.
Experiment using Torricelli's apparatus
Objective: measure the distance the flow goes
before hitting the ground (Ax) and relate this
to the height of the fluid column (ft = y^ — 2/1).
Data points for Ax and h are collected and a
best fit power law curve is graphed. Assume
that Ax and h are related by a power law given
by
V
Ax = a ft]
n
(12)
h
where n is to be determined.
By taking the natural logarithm of the data
points and graphing the relationship we should
find that the slope of the graph is the power of
the relationship. That is
U[$
X
u[W
A [In Ax]
A [In ft]
= n = slope
(13)
10
Theoretical Prediction
We know that
Ax
v = (14)
At V J
The distance m is related to the time it takes
the drop to fall by 2/1 = §#t 2 , so then
v = Ax
9
2j/i
= \/[2 £ h] (15)
where h = y 2 — 2/1 ■ This means that in theory
Ax = 2[yi h]2 (16)
where 2/1 is the height of the spigot above the
ground. Theory therefore tells us that the slope
of the datapoints on the log-log Ax and h graph
should be equal to i
1 1